# Prove that pivot columns of matrix A form a basis for C(A).

Prove that pivot columns of matrix A form a basis for C(A).

From "The Four Fundamental Subspaces".

Please show each step of your solution.

https://brainmass.com/math/linear-algebra/prove-pivot-columns-matrix-form-basis-171729

#### Solution Preview

*Prove that pivot columns of matrix A form a basis for C(A).

(The material is from THE FOUR FUNDAMENTAL SUBSPACES. Please show each step of your solution. Thank you.)

Proof. First of all, we need the following facts

Fact 1: The dimensions of C(A) and R(A) are equal, since both are equal to the rank of A;

Fact 2:

A matrix A is in Reduced Row-Echelon Form (RREF) if it has the following properties:

1) If a row does not consist entirely of zeroes, then the first nonzero entry in the row is a 1.

2) Any rows that consist entirely of zeroes are at the bottom of the matrix, below all nonzero rows.

3) For rows not consisting of all zeroes, ...

#### Solution Summary

It is proven that pivot columns of matrix A form a basis for C(A).